Weak Morse inequalities represent constraints on the number of generators of a chain complex \(C_*\) , which can be improved by replacing \(C_*\)  by chain complex quasi-isomorphic to it with less generators.
This strategy is used, for example, in [1]} where, endowing a regular cell complex \(X\)  with a discrete Morse function \(f\) , \(C_*(X)\)  is shown to be quasi-isomorphic to the Morse complex containing only the critical cells of \(f\) . Thus, \(c_q\)  can be taken to coincide with the number of critical cells of \(f\)  with dimension \(q\) . Similarly, in the case of a PL Morse function defined on a simplicial complex, strong and weak inequalities hold with \(c_q\)  being the number of critical vertices of index \(q\)  [2]}.
